Changing selection around a mesh object
First off, there is no way to rotate or move the area where the selection is. The selection is selecting something whether it be objects, vertices, edges, or faces, you have to select something.

Going from your picture, and using the view in that picture; selecting the faces on the other side will be easy.


Change to wireframe view, in the 3D view header, or by pressing Z.
Use Border Select B to select the same area that is already selected. However in now wireframe both sides will get selected.
Change the view back to solid view, again by pressing Z. Now use border select, but instead of pressing with the LMB , use the MMB . That will unselect everything that is inside of the box.